Asymptotic analysis of the Berry curvature in theE⊗eJahn-Teller model

Abstract

The effective Hamiltonian for the linear E ⊗ e Jahn-Teller model describes the coupling between two electronic states and two vibrational modes in molecules or bulk crystal impurities. While in the Born-Oppenheimer approximation the Berry curvature has a delta function singularity at the conical intersection of the potential energy surfaces, the exact Berry curvature is a smooth peaked function. Numerical calculations revealed that the characteristic width of the peak is K 1/2 /gM 1/2 , where M is the mass associated with the relevant nuclear coordinates, K is the effective internuclear spring constant and g is the electronic-vibrational coupling. This result is confirmed here by an asymptotic analysis of the M → ∞ limit, an interesting outcome of which is the emergence of a separation of length scales. Being based on the exact electron-nuclear factorization, our analysis does not make any reference to adiabatic potential energy surfaces or nonadiabatic couplings. It is also shown that the Ham reduction factors for the model can be derived from the exact geometric phase.

Regularized Born-Oppenheimer molecular dynamics

“…On the other hand, a nonzero value of Planck's constant (no matter how small) leads to regular nuclear trajectories evolving under the influence of smoothened effective potentials given by the convolution of the original potentials with the nuclear probability density. In turn, the removal of the singularities makes the Berry phase a geometric (path-dependent) quantity in analogy with the recent findings in [44,43]. As we shall see in the simple case of Jahn-Teller systems, the geometric phase resulting from the Bohmian closure successfully reproduces the phenomenology appearing in exact nonadiabatic studies [43]: starting from zero, the Berry phase rapidly tends to the Aharonov-Bohm topological index as the loop encircling the conical intersection becomes distant from the singularity.…”

“…has an intrinsic topological nature and it is actually expressed as a topological index. While this topological character is well established in the standard Aharonov-Bohm effect [3], its manifestation has recently been debated in [39,44,43]. Therein, Gross and collaborators have shown that the molecular Berry phase may be considered as a manifestation of a geometric (path-dependent) effect.…”

“…Indeed, notice that even if ζ has a pole at the location of the conical intersection, the geometric phase Γ vanishes as the loop γ 0 encircling the singular point of ζ shrinks to the point itself. The emergence of this geometric type of Berry phase has recently been considered in the context of quantum chemistry by Gross and collaborators [43,44]. In some specific cases, the authors of [43,44] adopt an exact approach [1] to the full nonadiabatic quantum molecular problem to show that the exact Berry connection quickly tends to the topological value¸γ 0 ∇ζ(r) • dr from Born-Oppenheimer theory as the loop γ 0 encircling the singular point of ζ becomes larger and larger.…”

“…The emergence of this geometric type of Berry phase has recently been considered in the context of quantum chemistry by Gross and collaborators [43,44]. In some specific cases, the authors of [43,44] adopt an exact approach [1] to the full nonadiabatic quantum molecular problem to show that the exact Berry connection quickly tends to the topological value¸γ 0 ∇ζ(r) • dr from Born-Oppenheimer theory as the loop γ 0 encircling the singular point of ζ becomes larger and larger. According to the authors of [44], this feature would explain why Born-Oppenheimer theory is so successful in reproducing the values of the Berry phase observed in molecular spectroscopy experiments [12,13,54].…”

When the exact factorization meets conical intersections...

“…This study was the first detailed analysis in the time domain of the properties of the EF in connection to CIs. (The EF can also be performed in the time-independent picture [28][29][30][31][32][33][34][35], and analyses of CI in this framework were also proposed [36][37][38][39]. )…”